# Binary numbers

To understand the binary numbering system, it may be useful to understand how numbering systems work. You are likely familiar with the base-10 number system since it is the basis for the universally used decimal number system. It is taught in elementary school. The following is a description of base-10 numbers followed by a description of base-2 numbers (binary).

#### Base-10 number system

Numbers are most commonly described using the decimal system, which is a base-10 system. A base-10 system is so-called because there are 10 different symbols used to represent any number. The base-10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. To represent a larger number than 9, additional digits are required. Each digit has a multiplier value applied to it's symbol's value that is 10 times larger than the multiplier of the digit on its right. The value of the multiplier for the $n^{th}$ digit for a base-10 system is $10^{n-1}$. The first digit on the right has the face value of the its symbol. If a second digit from the right is present, it's value is 10 times its symbol's value. A third digit would have a value 100 times its symbol's value.

To represent the value $98$, two symbols are required. The second symbol from the right has multiplier of $10^1$ or $10$ and so a value of $9x10^1 = 90$. The first symbol has a multiplier of $10^0$ or $1$ and a value of $8x10^0 = 8$. The value of the number is the sum of the value of its digits. For 98, the value is $90 + 8$. A 4 digit base-10 example is provided in Figure 1.

Figure 1: How the value of a base-10 number is determined.

#### Base-2 number system

The numbering system of interest here is the base-2, or binary. The $n^{th}$ consecutive digit has a multiplier value equal to $2^{n-1}$. The binary number $1010$ has a base-10 value of 10 as demonstrated in Figure 2. The first digit has a value of $2^0\times 0$, the second digit has a value of $2^1 \times 1 = 2$, the third digit has a value of $2^2\times 0=0$ and the fourth digit has a value of $2^3\times1=8$. Summing the values of the digits results in $8+0+2+0 = 10$.

Figure 2: How the value of a base-2 number is determined.